Chapter 4, Problem 108P

The velocity field for an incompressible flow is given as $\stackrel{\to }{V}=5x^2\stackrel{\to }{i}-20xy\stackrel{\to }{j}+100t\stackrel{\to }{k}$. Determine if this flow is steady. Also determine the velocity and acceleration of a particle at (1, 3, 3) at: =0.2 s.

To determine

The velocity of particle.

The acceleration of particle

Answer to Problem 108P

The velocity of particle is $\underset{\_}{\stackrel{\rightharpoonup }{V}=5\stackrel{\rightharpoonup }{i}-60\stackrel{\rightharpoonup }{j}+20\stackrel{\rightharpoonup }{k}}$.

The acceleration of particle is $\underset{\_}{\stackrel{\rightharpoonup }{a}=50\stackrel{\rightharpoonup }{i}+900\stackrel{\rightharpoonup }{j}+100\stackrel{\rightharpoonup }{k}}$.

Explanation of Solution

The vector field of flow is $\stackrel{\rightharpoonup }{V}=5x^2\stackrel{\rightharpoonup }{i}-20xy\stackrel{\rightharpoonup }{j}+100t\stackrel{\rightharpoonup }{k}$, and the position of particle is $\left(1,3,3\right)$ at time equal to $0.2\text{s}$

Write the expression for the velocity field vector.

  $\stackrel{\rightharpoonup }{V}=u\stackrel{\rightharpoonup }{i}+v\stackrel{\rightharpoonup }{j}+w\stackrel{\rightharpoonup }{k}$

Here, the velocity in $x$ direction is $u$, the velocity in $y$ direction is $v$ and the velocity in $z$ direction is $w$.

Write the expression for the given velocity field vector.

  $\stackrel{\rightharpoonup }{V}=5x^2\stackrel{\rightharpoonup }{i}-20xy\stackrel{\rightharpoonup }{j}+100t\stackrel{\rightharpoonup }{k}$   ...... (I)

Here, the velocity in $x$ direction is $5x^2$, the velocity in $y$ direction is $-20xy$ and the velocity in $z$ direction is $100t$.

Write the expression for the acceleration component in $x$ direction.

  $a_x=\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}$   ...... (II)

Write the expression for the acceleration component in $y$ direction.

  $a_y=\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}$   ...... (III)

Write the expression for the acceleration component in $z$ direction.

  $a_z=\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}$   ...... (IV)

Substitute $5x^2$ for $u$, $-20xy$ for $v$ and $100t$ for $w$ in Equation (II).

  $\begin{array}{c}{a}_x=\frac{\partial \left(5x^2\right)}{\partial t}+5x^2\frac{\partial \left(5x^2\right)}{\partial x}+\left(-20xy\right)\frac{\partial \left(5x^2\right)}{\partial y}+100t\frac{\partial \left(5x^2\right)}{\partial z}\\ =0+50x^3-0+0\\ =50x^3\end{array}$..... (V)

Substitute $5x^2$ for $u$, $-20xy$ for $v$ and $100t$ for $w$ in Equation (III).

  $\begin{array}{c}{a}_y=\frac{\partial \left(-20xy\right)}{\partial t}+5x^2\frac{\partial \left(-20xy\right)}{\partial x}+\left(-20xy\right)\frac{\partial \left(-20xy\right)}{\partial y}+100t\frac{\partial \left(-20xy\right)}{\partial z}\\ =0-100x^{2}y+400x^{2}y+0\\ =300x^{2}y\end{array}$..... (VI)

Substitute $5x^2$ for $u$, $-20xy$ for $v$ and $100t$ for $w$ in Equation (IV).

  $\begin{array}{c}{a}_z=\frac{\partial \left(100t\right)}{\partial t}+5x^2\frac{\partial \left(100t\right)}{\partial x}+\left(-20xy\right)\frac{\partial \left(100t\right)}{\partial y}+100t\frac{\partial \left(100t\right)}{\partial z}\\ =100+0+0+0\\ =100\end{array}$..... (VII)

Write the expression for acceleration vector.

  $\stackrel{\rightharpoonup }{a}=a_x\stackrel{\rightharpoonup }{i}+a_y\stackrel{\rightharpoonup }{j}+a_z\stackrel{\rightharpoonup }{k}$   ...... (VIII)

Calculation:

Substitute $1$ for $x$, $3$ for $y$ and $0.2\text{s}$ for $t$ in Equation (I).

  $\begin{array}{c}\stackrel{\rightharpoonup }{V}=5\text{×}{\left(1\right)}^2\stackrel{\rightharpoonup }{i}-\left(20\text{×1×2}\right)\stackrel{\rightharpoonup }{j}+\left(100\text{×0}\text{.2}\text{s}\right)\stackrel{\rightharpoonup }{k}\\ \stackrel{\rightharpoonup }{V}=5\stackrel{\rightharpoonup }{i}-60\stackrel{\rightharpoonup }{j}+20\stackrel{\rightharpoonup }{k}\end{array}$

Substitute $1$ for $x$ in Equation (V).

  $\begin{array}{c}{a}_x=50{\left(1\right)}^3\\ =50\end{array}$

Substitute $1$ for $x$ and $3$ for $y$ in Equation (VI).

  $\begin{array}{c}{a}_y=300{\left(1\right)}^2\left(3\right)\\ =900\end{array}$

Substitute $900$ for $a_y$, $50$ for $a_x$ and $100$ for $a_z$ in Equation (VIII).

  $\stackrel{\rightharpoonup }{a}=50\stackrel{\rightharpoonup }{i}+900\stackrel{\rightharpoonup }{j}+100\stackrel{\rightharpoonup }{k}$

Conclusion:

The answer is calculated by by the expression : $a_y=\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}$